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Infinite fuzzy computations
Fuzzy Sets and Systems 153 (2005) 275 – 288 www.elsevier.com/locate/fss
Infinite fuzzy computations George Rahonis∗ Department of Mathematics, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece Received 19 May 2004; received in revised form 8 December 2004; accepted 12 December 2004 Available online 6 January 2005
Abstract We consider Büchi automata weighted over the fuzzy semiring F = ([0, 1], max, min, 0, 1). A Büchi theorem for the class of -recognizable series with coefficients in F, i.e., for behaviors of Büchi automata over F is established. Furthermore, we show that an infinitary language is Büchi recognizable iff its characteristic series is -recognizable. © 2004 Elsevier B.V. All rights reserved. Keywords: Fuzzy system models; Formal series; Infinite words
1. Introduction In 1961, Schützenberger [22] introduced the concept of a weighted automaton which is in fact a classical automaton M (over an alphabet ) whose transitions are equipped with a weight over a semiring S = (S, ⊕, , 0, 1). Then to each input word u and each run of M at u, a value from S is assigned which is induced by multiplying the weights of the transitions consisting the run. Furthermore, the weight of u is obtained by summing up the weights of all successful runs of M on u. In this way the behavior of M is a formal power series |M| : ∗ → S which is called recognizable. The classical Schützenberger theorem states that Re c(, S) = Rat(, S), ∗ Tel.: +30 2310 998330; fax: +30 2310 997983.
E-mail address: [email protected] (G. Rahonis). 0165-0114/$ - see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2004.12.003
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i.e., the class of recognizable formal power series over and S equals that of rational power series. Since then a large number of papers appeared in the literature, concerning weighted automata over several semirings, and formal power series contributed to many applications in Computer Science (for extended lists of references see [3,8,11,10]). On the other hand, automata over infinite objects can serve as models of infinite processes like operating systems, web servers applications, etc. Furthermore, they play a crucial role in monadic second-order logic and game theory. One of the most important machines over infinite words is the Büchi automaton. Intuitively, a Büchi automaton A over an alphabet accepts a finite word u in the way a classical automaton does, whereas it accepts an infinite word w if there exists an infinite run of A on w which starts at the initial state and meets at least one -final state infinitely often. A classical result of Büchi states that − Re c() = − Rat(),
i.e., the classes of all Büchi recognizable and -rational languages over coincide. In the last few years formal power series and weighted automata over infinite words have attracted the interest of researchers. In this case, convergence problems may arise according to the operations of the underlying semiring. The first effort to this direction has been made in [5]. In that paper Culik II and Karhumäki considered weighted automata over R (the reals) computing real functions. Since convergence problems arose in R, the authors restricted themselves to the special class of level automata. Krithivisan and Sharda introduced in [9] fuzzy -automata considering various modes of acceptance. Their fuzzy -automata with Büchi condition are in fact fuzzy Büchi automata in our sense, restricted on infinite words. Recently in [6] (see also [7]), a Büchi theorem is proved for infinitary power series over the semiring Rmax = (R 0 ∪ {−∞}, max, +, −∞, 0). In that case, convergence handicaps are dealt with by considering the skew product of power series instead of the classical one. In this paper we deal with formal power series over infinite words with coefficients in the fuzzy semiring F = ([0, 1], max, min, 0, 1). The -rational operations among infinitary series are max −, min −, ∗ −, and -closure. − Rat(, F ) stands for the class of -rational series over an alphabet with coefficients in F . We introduce the notion of fuzzy Büchi automata which are Büchi automata weighted over F . Our main result, an extension of Büchi theorem, states that − Re c(, F ) = − Rat(, F ),
where − Re c(, F ) is the set of series obtained as behaviors of fuzzy Büchi automata. A similar result for finite words has been proved by Santos in [21]. In that paper, the author used the so-called maximin automata [20] which are in fact weighted automata over F . In the framework of formal series theory, the result of Santos is a classical Schützenberger theorem [22]. In our paper, no convergence shortcomings appear within the fuzzy semiring, due to the max and min operations. The notion of the fuzzy Büchi automaton is the natural extension of the classical one, in the framework of formal series over F . Precisely, a Büchi automaton is a fuzzy Büchi automaton with all its weights equal to 1. The structure of the paper is the following: In Section 2, we present some preliminary matter concerning finitary series over F . We restate the result of Santos [21] in terms of formal series theory. Section 3 is devoted to infinitary series. We introduce the notion of fuzzy Büchi automata and we state an effective
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normalization procedure. We establish the Büchi theorem for the class of -recognizable series, i.e., series obtained as behavior of fuzzy Büchi automata. Next, we show that -recognizability is preserved by strict homomorphisms. Finally, we prove that a language is Büchi recognizable iff its characteristic series is -recognizable. 2. Finitary formal series A semiring A = (A, +, ·, 0, 1) consists of a set A endowed with two binary operations + and · and two constant elements 0 and 1, such that: (i) (ii) (iii) (iv)
(A, +, 0) is a commutative monoid, (A, ·, 1) is a monoid, a · (b + c) = a · b + a · c and (a + b) · c = a · c + b · c, for each a, b, c ∈ A (distributivity law), a · 0 = 0 = 0 · a for each a ∈ A.
Further A is called commutative if a · b = b · a for all a, b ∈ A. The interval F = [0, 1] with the max and min operations obtains the semiring structure. We call this semiring the fuzzy semiring (this name first appeared in [19]), and we denote it by F , i.e. F = (F, max, min, 0, 1).
Obviously F is commutative. In the sequel, we concentrate on infinite subsets of F ; thus, we use sup (least upper bound) instead of max, and inf (greatest lower bound) instead of min. We shall denote by sup A (resp. inf A) the least upper bound (resp. the greatest lower bound) of a set A. will stand for a finite alphabet and ∗ for the free monoid generated by , i.e., for the set of all finite words over . Let w = x0 x1 . . . xn−1 ∈ ∗ , with x0 , . . . , xn−1 ∈ . We shall use the notation w = w(0)w(1) . . . w(n − 1) where w(i) = xi , for i = 0, . . . , n − 1. will denote the empty word. Then + = ∗ − {}. A mapping S : ∗ → F is called formal power series over and F . The values S(w), w ∈ ∗ are usually written as (S, w) and referred to as the coefficients of the series. F ∗ denotes the set of all series over with coefficients in F . It is well-known that F ∗ inherits the semiring structure [3,11,10]. For S, T ∈ F ∗ the addition S ⊕ T : ∗ → F is defined by (S ⊕ T , w) = sup{(S, w), (T , w)},
for w ∈ ∗ ,
whereas the multiplication S T : ∗ → F is determined by (S T , w) = sup{inf{(S, w1 ), (T , w2 )}/w = w1 w2 },
for w ∈ ∗ .
The neutral elements for the above operations are, respectively, 0 : ∗ → F, and ∗
1 : → F,
for w ∈ ∗
(0, w) = 0 (1, w) =
1 0
if w = , else.
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The support of a series S ∈ F ∗ is the language sup p(S) = {w | w ∈ ∗ and (S, w) = 0}. A series with finite support is called polynomial. As usual F ∗ denotes the set of all polynomials over and F . Let a ∈ F and S ∈ F ∗ . The scalar product of a with S is the series aS : ∗ → F given by (aS, w) = inf{a, (S, w)}
for each w ∈ ∗ .
Then a series S : ∗ → F can be written as a formal sum S = sup{(S, w)w | w ∈ ∗ }, where for each w ∈ ∗ the series w : ∗ → F is defined by 1 if u = w, (w, u) = 0 else S will be called proper if (S, ) = 0. For S ∈ F ∗ and n 0, the consecutive powers S n of S are inductively determined by S 0 = 1,
S n+1 = S n S,
n 0.
It is clear that for n 1 and w ∈ ∗ , it holds (S n , w) = sup{inf{(S, w1 ), . . . , (S, wn )} | w = w1 . . . wn }. Further for S proper, the series S ∗ : ∗ → F is defined to be S ∗ = sup S n , i.e., (S ∗ , w) = sup{(S n , w) | n 0}. Obviously, (S n , w) = 0, for n > |w|, where |w| denotes the length of the word w. The rational operations of formal power series are ⊕−, − and ∗ −. Rat(, F ) is the least set of series in F ∗ containing the polynomials and closed under the rational operations. The elements of Rat(, F ) are named rational series. Remark. The reader may observe that we should define the ∗ −operation (as well as the -operation later) on each series, because of the completeness property of the fuzzy semiring (see [10]). Nevertheless, we restrict ourselves to proper series for technical reasons only (see Lemma 6 below). Obviously, this fact does not lighten our Büchi theorem. We now give the definition of fuzzy weighted automata. They are fuzzy automata weighted over F . In [20,21] such automata are called maximin automata. Further, fuzzy automata and fuzzy automata weighted over semirings have been extensively studied by many authors (see for e.g. [1,2,4, 12–18,24–28]).
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Definition 1. A fuzzy weighted automaton is a system A = (Q, , , f, init, ter) where • • • • • •
Q is the finite set of states, is the finite alphabet of inputs, ⊆ Q × × Q is the set of transitions, f : → F is a mapping assigning to each transition a weight from F,
init : Q → F is the initial distribution, and ter : Q → F is the terminal distribution.
Let w = w(0) . . . w(n − 1) ∈ ∗ . A run of A at w is a word rw ∈ Q∗ , rw = rw (0) . . . rw (n), rw (0), . . . , rw (n) ∈ Q, such that (rw (i), w(i), rw (i + 1)) ∈ , for each i = 0, . . . , n − 1. The weight of the run rw is inf{f (rw (i), w(i), rw (i + 1)) | i = 0, . . . , n − 1}. We define that the weight of the empty run i.e., the run on the empty word, equals 1. The behavior of A is the series |A| : ∗ → F determined by (|A|, w) = sup{inf{init(rw (0)), inf{f (rw (i), w(i), rw (i + 1)) | i = 0, . . . , n − 1}, ter(rw (n))} | rw is a run of A at w} for all w ∈ ∗ . Observe that (|A|, ) = 0 if there exists at least one state q ∈ Q, such that init(q) = 0 and ter(q) = 0. Since then, we assume that there exists an empty run r from q to q, with weight inf{init(q), ter(q)}. A formal power series S ∈ F ∗ is called recognizable if there exists a fuzzy weighted automaton A, with S = |A|. The set of all recognizable series in F ∗ is denoted by Re c(, F ). Theorem 2 (Santos [21], Schützenberger [22]). Re c(, F ) = Rat(, F ). 3. Infinitary series For a finite alphabet , we denote by the set of all infinite (or -words) over . An infinite word w ∈ is written in the form w = w(0)w(1)w(2) . . . , with w(i) ∈ , i = 0, 1, 2, . . . . The set of all words over is ∞ , i.e., ∞ = ∗ ∪ . The concatenation of two words in ∞ is defined in the following way: w1 w2 if w1 ∈ ∗ w1 w2 = w1 if w1 ∈ . An infinitary series over and F is a mapping S : ∞ → F . The set of all infinitary series over and F is denoted by F ∞ .
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Example 3. Let L ⊆ ∞ . The characteristic series of L, char L : ∞ → F is given by 1 if w ∈ L, (char L , w) = 0 else. A series S ∈ F ∞ can be written as a “sum" S = Sfin ⊕ Sinf with Sfin : ∗ → F and Sinf : → F , given, respectively, by (S, w), w ∈ ∗ , (Sfin , w) = 0 else and
(Sinf , w) =
(S, w), 0
w ∈ , else.
Consider S, T ∈ F ∞ . Their sum and product are the infinitary series S ⊕ T , S T whose coefficients are determined, respectively, by the following clauses: (S ⊕ T , w) = sup{(S, w), (T , w)}
for each w ∈ ∞ ,
and (S T , w) = sup{inf{(S, w1 ), (T , w2 )} | w = w1 w2 }
for each w ∈ ∗ ,
whereas (S T , w) = sup{(S, w), sup{inf{(S, w1 ), (T , w2 )} | w = w1 w2 , w1 ∈ ∗ }},
for each w ∈ .
Further the -closure of a proper finitary series S is the infinitary series S : → F with coefficients (S , w) = sup{inf{(S, w1 ), (S, w2 ), . . .} | w = w1 w2 . . . , with w1 , w2 , . . . ∈ ∗ }, for each w ∈ . The -rational operations in F ∞ are ⊕−, −, ∗ − and -closure, where ∗ − and -closure are defined for proper finitary series. − Rat(, F ) stands for the least class of series in F ∞ containing the polynomials and closed under the -rational operations. Its elements are termed -rational series. In the sequel, we introduce the notion of fuzzy Büchi automata. They are Büchi automata weighted over F . Such automata restricted on infinite words are called fuzzy -automata with Büchi condition in [9]. Definition 4. A fuzzy Büchi automaton (FBA) is a 7-tuple A = (Q, , , f, init, ter, ter ) where • Q is the finite state set, • is the finite input alphabet,
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• • • • •
281
⊆ Q × × Q is the set of transitions, f : → F is a mapping assigning to each transition a weight from F, init : Q → F is the initial distribution, ter : Q → F is the terminal distribution, and ter : Q → F is the -terminal distribution.
Consider a word w = w(0) . . . w(n−1) ∈ ∗ . A run of A at w is a word rw ∈ Q∗ , rw = rw (0) . . . rw (n), rw (0), . . . , rw (n) ∈ Q, such that (rw (i), w(i), rw (i + 1)) ∈ , for each i = 0, . . . , n − 1. The weight of the run rw is inf{f (rw (i), w(i), rw (i + 1)) | i = 0, . . . , n − 1}. The finitary behavior of A is the series |A|fin : ∗ → F with coefficients (|A|fin , w) = sup{inf{init(rw (0)), inf{f (rw (i), w(i), rw (i + 1)) | i = 0, . . . , n − 1}, ter(rw (n))} | rw is a run of A at w} for all w ∈ ∗ . Observe that the finitary behavior of A is the behavior of the fuzzy weighted automaton (Q, , , f, init, ter). Now let w = w(0)w(1) . . . be an infinite word over . A run of A at w is an infinite word rw = rw (0)rw (1) . . . ∈ Q such that (rw (i), w(i), rw (i + 1)) ∈ , for all i = 0, 1, . . . . Assume J ⊆ N to be an infinite index set (where N is the set of naturals). The weight of the run rw with terminals J is defined to be the number rwJ = inf{init(rw (0)), inf{f (rw (i), w(i), rw (i + 1)) | i = 0, 1, . . .}, inf{ter (rw (j )) | j ∈ J }}. The infinitary behavior of A is the series |A|inf : → F given by (|A|inf , w) = sup{rwJ | rw is a run of A at w, J ⊆ N is infinite} for each w ∈ . The behavior of A is the series | A | : ∞ → F defined by |A| = |A|fin ⊕ |A|inf . A series S ∈ F ∞ is termed to be -recognizable if there exists an FBA A, such that S = |A|. We denote by − Re c(, F ) the class of all -recognizable series over the alphabet .
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Definition 5. An FBA A = (Q, , , f, init, ter, ter ) is called normalized if there are two states q0 , qt ∈ Q such that 1 if q = qt , 1 if q = q0 , ter(q) = init(q) = 0 else, 0 else and there are no transitions of the forms (q, a, q0 ) and (qt , a, q) in . We call q0 and qt the initial and terminal state, respectively. Such an automaton will be denoted by A = (Q, , , f, q0 , qt , ter ). Further, an FBA A = (Q, , , f, init, ter, ter ) is termed weak normalized if it has an initial state q0 as above. In this case we write A = (Q, , , f, q0 , ter, ter ). We establish the following lemma. Lemma 6. Let A = (Q, , , f, init, ter, ter ) be an FBA. We can effectively construct a normalized FBA A such that sup p(|A |fin ) = sup p(|A|fin ) ∩ + and |A |inf = |A|inf . Further, we can effectively construct a weak normalized FBA A such that |A | = |A|. Proof. Consider the new states q0 , qt and put Q = Q ∪ {q0 , qt }. Let A = (Q , , , f , q0 , qt , ter ) be the normalized FBA with set of transitions = ∪ {(q0 , a, q ) | ∃q ∈ Q : (q, a, q ) ∈ } ∪ {(q, a, qt ) | ∃q ∈ Q : (q, a, q ) ∈ } ∪ {(q0 , a, qt ) | ∃q, q ∈ Q : (q, a, q ) ∈ }.
The weight function f : → F is given by the following clauses: • • • •
f (q, a, q ) = f (q, a, q ), ∀(q, a, q ) ∈ , f (q0 , a, q ) = sup{inf{init(q), f (q, a, q )} | (q, a, q ) ∈ }, ∈ , q ∈ Q, f (q, a, qt ) = sup{inf{f (q, a, q ), ter(q )} | (q, a, q ) ∈ }, ∈ , q ∈ Q, f (q0 , a, qt ) = sup{inf{init(q), f (q, a, q ), ter(q )} | (q, a, q ) ∈ }, ∈ .
The reader will have no difficulties in verifying that the FBA A has the announced properties. The weak normalization procedure of A is similar to the previous one. Now we are ready to prove our main result. Theorem 7. A series S ∈ F ∞ is -rational iff it is -recognizable. Proof. We establish the proof in two steps. Step 1. − Rat(, F ) ⊆ − Re c(, F ). It suffices to prove that the class − Re c(, F ) contains the polynomials and it is closed under the -rational operations. Because of Theorem 2, Rat(, F ) ⊆ − Re c(, F ) and thus the polynomials are -recognizable. Let S ∈ − Re c(, F ) be a proper finitary series i.e. S ∈ Rec(, F ). Then once again by Theorem 2, S ∗ ∈ Re c(, F ), and so S ∗ ∈ − Re c(, F ). Further, there exists a normalized FBA A =
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(Q, , , f, q0 , qt , 0) such that |A| = S. We identify the states q0 and qt , and we consider the FBA A = (Q, , , f, q0 , 0, ter )
with ter (q)
=
1 0
if q = q0 , else.
Let w = w(0)w(1) . . . be an infinite word in , rw be a run of A at w, and J ⊆ N an infinite set, namely J = {j1 , j2 , . . .}, with j1 < j2 < · · · . Then rwJ = inf{init(rw (0)), inf{f (rw (i), w(i), rw (i + 1)) | i = 0, 1, . . .}, inf{ter (rw (j )) | j ∈ J }}. Assume that rwJ = 0. Then rw (0) = q0 and rw (j ) = q0 , ∀j ∈ J . Thus rwJ = inf{inf{inf{f (rw (i), w(i), rw (i + 1)) | i = jk , jk + 1, . . . , jk+1 − 1} | k = 0, 1, . . .}, inf{ter (rw (jk )) | k = 0, 1, . . .}} = inf{inf{f (rw (i), w(i), rw (i + 1)) | i = jk , jk + 1, . . . , jk+1 − 1} | k = 0, 1, . . .}, where j0 = 0. That is rwJ = inf{(S, w(0) . . . w(j1 − 1)), (S, w(j1 ) . . . w(j2 − 1)), . . .}. Conversely, by similar arguments we can show that for each decomposition w = w1 w2 . . . , w1 , w2 . . . ∈
∗ of w ∈ , we can find an infinite J ⊆ N and a run rw of A on w, such that
rwJ = inf{(S, w1 ), (S, w2 ), . . .}. We conclude that |A | = S and so S ∈ − Re c(, F ). Now let S, T ∈ − Re c(, F ) and A1 = (Q1 , , 1 , f1 , init 1 , ter 1 , ter 1 ), A2 = (Q2 , , 2 , f2 , init 2 , ter 2 , ter 2 ) be two FBA’s with |A1 | = S and |A2 | = T . Without any loss, we can assume that Q1 ∩ Q2 = ⭋. We construct the FBA A = (Q, , , f, init, ter, ter )
whose components are given by • Q = Q1 ∪ Q2, = 1 ∪ 2 , f1 (q, a, q ) if (q, a, q ) ∈ 1 , • f (q, a, q ) = f2 (q, a, q ) if (q, a, q ) ∈ 2 , init 1 (q) if q ∈ Q1 , • init(q) = init 2 (q) if q ∈ Q2 , ter 1 (q) if q ∈ Q1 , • ter(q) = ter (q) if q ∈ Q2 , 2 ter 1 (q) if q ∈ Q1 , • ter (q) = ter 2 (q) if q ∈ Q2 . It is a tedious task to formally state that |A| = S ⊕ T .
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Further, let S, T ∈ − Re c(, F ) with sup p(S) ⊆ ∗ and sup p(T ) ⊆ . Let A1 = (Q1 , , 1 , f1 , q01 , qt1 , 0) be a normalized FBA with |A1 | = S, and A2 = (Q2 , , 2 , f2 , q02 , 0, ter 2 ) be a weak normalized FBA with |A2 | = T . Again we assume that Q1 ∩ Q2 = ∅. We consider the FBA, A = (Q, , , f, q01 , 0, ter 2 ),
where Q, and f are determined as in the above construction, and we have identified the states qt1 and q02 . The reader will have no difficulties in showing that |A| = S T . In order to complete the proof of Step 1, we have to establish that S T ∈ − Re c(, F ) for arbitrary -recognizable series S, T . But S T = (Sfin ⊕ Sinf ) (Tfin ⊕ Tinf ) and because of the distributivity law in F S T = (Sfin Tfin ) ⊕ (Sfin Tinf ) ⊕ (Sinf Tfin ) ⊕ (Sinf Tinf ). By the definition of the -operation we have (Sinf Tfin ) = (Sinf Tinf ) = Sinf and thus S T = (Sfin Tfin ) ⊕ (Sfin Tinf ) ⊕ Sinf which by the previous arguments gives S T ∈ − Re c(, F ) as wanted. Step 2. − Re c(, F ) ⊆ − Rat(, F ). Let S ∈ − Re c(, F ). Without any loss we can assume S to be proper. There exists a normalized FBA A = (Q, , , f, q0 , qt , ter ) with |A| = S. Consider the FBA’s, A1 = (Q, , , f, q0 , qt , 0) (obviously |A1 | = |A1 |fin = Sfin ), A2 = (Q, , , f, q0 , ter 2 , 0) with ter 2 = ter , A3 = (Q, , , f, init 3 , ter 3 , 0) with init 3 = ter 3 = ter .
It is not hard to understand that the support of A2 is composed of all the finite prefixes of infinite words in the support of A. Further, Sinf = |A2 | |A3 | = |A2 |fin (|A3 |fin ) . Thus, S = |A1 |fin ⊕ (|A2 |fin (|A3 |fin ) ). Since |A1 |fin , |A2 |fin , |A3 |fin are rational power series (Theorem 2) S is -rational and this completes the proof. Next we show that strict homomorphisms preserve -recognizability.
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285
Let , be finite alphabets and h : ∗ → ∗ be a homomorphism. Recall that h is called strict if h(a) ∈ + , for all a ∈ . h is extended to infinite words in the obvious way: If w = w(0)w(1)... ∈ , then h(w) = h(w(0))h(w(1)).... Furthermore, h is extended to a mapping −
h : F ∞ → F ∞
by the formula −
h(S) = sup{(S, w)h(w) | w ∈ ∞ }
for all S ∈ F ∞ .
Proposition 8. Assume S ∈ F ∞ to be -recognizable and h : ∗ → ∗ to be a strict homomor−
phism. Then the series h(S) ∈ − Re c(, F ). Proof. Let A = (Q, , , f, q0 , ter, ter ) be a weak normalized FBA with behavior S. For each a ∈ , such that h(a) = 1 . . . n , i ∈ , i = 1, . . . , n, and each transition (q, a, q ) ∈ , we consider the new states p1 , . . . , pn−1 not belonging to Q, and the new transitions (q, 1 , p1 ), (p1 , 2 , p2 ), . . . , (pn−1 , n , q ). Let P be the set of the new states obtained in the above way, and the set of the new transitions. Let also A = (Q , , , f , q0 , ter , ter )
be the weak normalized FBA with its components given by the following clauses: • Q = Q ∪ P , • f (q, 1 , p1 ) = f (q, a, q ) and f (p1 , 2 , p2 ) = · · · = f (pn−1 , n , q ) = 1, where the transitions (q, 1 , p1 ),(p1 , 2 , p2 ), . . . , (pn−1 , n , q ) obtained from (q, a, q ) in the above-described procedure, ter(q) if q ∈ Q, • ter (q) = 0 else, ter (q) if q ∈ Q, • ter (q) = 0 else. Assume now that w = w(0) . . . w(n − 1) ∈ ∗ and rw is a run of A at w. Obviously, there is a run rh(w) of A at h(w), with the same weight. Conversely, if ru is a run of A at u ∈ ∗ , then there exists a run rw of A at the word w, such that h(w) = u, and the weights of ru and rw are equal. This means that −
h(S)fin = |A |fin . Further, let w ∈ , rw be a run of A at w and J ⊆ N be an infinite index set such that rwJ = 0. From the construction of the automaton A , there is a run rh(w) of A at h(w) and an infinite index set J ⊆ N
J . On the other hand, if r is a run of A at u ∈ , and J ⊆ N is an infinite index such that rwJ = rh(w) u
set such that ruJ = 0, then there clearly exists a word w ∈ , with h(w) = u, a run rw of A at w, and
−
−
an infinite index set J ⊆ N such that rwJ = ruJ . Thus, h(S)inf = |A |inf . We conclude that h(S) = |A |, −
and so h(S) ∈ − Re c(, F ).
In the sequel, we investigate the relation among Büchi recognizable languages and -recognizable series. First, we recall the notion of a Büchi recognizable language [22].
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Definition 9. A Büchi automaton is a system A = (Q, , , q0 , T , T ), where • • • • • •
Q is the finite state set, is the input alphabet,
q0 ∈ Q is the initial state, ⊆ Q × × Q is the set of transitions, T ⊆ Q is the set of final states, and T ⊆ Q is the set of -final states.
A run of A on a finite word w = w(0) . . . w(n − 1) ∈ ∗ is a word rw = rw (0) . . . rw (n) ∈ Q∗ such that (rw (i), w(i), rw (i + 1)) ∈ , for each i = 0, 1, . . . , n − 1. rw is called successful if rw (0) = q0 and rw (n) ∈ T . A run of A on an infinite word w = w(0)w(1) . . . ∈ is an infinite word rw = rw (0)rw (1) . . . ∈ Q , such that (rw (i), w(i), rw (i + 1)) ∈ , for all i = 0, 1, . . . . An infinite run rw is called successful if rw (0) = q0 and at least one state of T occurs infinitely many often among the rw (i)’s. The language accepted by A is L(A) = {w | w ∈ ∞ , there exists a successful run of A at w}. L ⊆ ∞ is called Büchi recognizable if L = L(A) for some Büchi automaton A. First, we prove that Proposition 10. Let L ⊆ ∞ be a Büchi recognizable language. Then its characteristic series char L : ∞ → F is -recognizable. Proof. Assume A = (Q, , , q0 , T , T ) to be a Büchi automaton recognizing L. Consider the weak normalized FBA, −
A = (Q, , , f, q0 , ter, ter )
with f : → F given by f (q, a, q ) = 1, and
ter(q) =
1 0
for each (q, a, q ) ∈
if q ∈ T , else,
ter (q) =
1 0
if q ∈ T , else.
It is clear from the constructions that for a given word w ∈ ∞ a successful run of A at w exists iff
−
−
(| A |, w) = 1. Thus | A | = char L . We obtain that char L ∈ − Re c(, F ). Next we establish the inverse implication. Proposition 11. Let L ⊆ ∞ . If char L ∈ − Re c(, F ) then L is Büchi recognizable.
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Proof. Let A = (Q, , , f, q0 , ter, ter ) be a weak normalized FBA with behavior char L , i.e. w∈L
iff (|A|, w) = 1.
We construct the Büchi automaton −
A = (Q, , , q0 , T , T )
with • T = {q | q ∈ Q and ter(q) = 0}, and • T = {q | q ∈ Q and ter (q) = 0}. To each run rw of A on a word w ∈ ∞ , corresponds a run r¯ of A¯ on w and vice versa (in fact r¯ = rw ). If w = w(0) . . . w(n − 1) ∈ ∗ , then inf{init(rw (0)), inf{f (rw (i), w(i), rw (i + 1) | i = 0, . . . , n − 1}, ter(rw (n))} = 0 iff r¯ is a successful run of A¯ . Further, if w ∈ , and J ⊆ N is an infinite index set such that rwJ = 0, then r¯w is successful. The converse is also obvious. We conclude (|A|, w) = 1
iff w ∈ L(A¯ ),
i.e., (char L , w) = 1
iff w ∈ L(A¯ )
and so L = L(A¯ ) which means that L is a Büchi recognizable language. 4. Conclusion We have considered Büchi automata weighted over the fuzzy semiring and we established a normalization procedure for them. We showed that their behaviors called -recognizable series constitute the class of -rational series over F . Furthermore, we proved that a string language is Büchi recognizable iff its characteristic series is -recognizable. Hopefully, Büchi automata over F can stand as models of infinite processes under a fuzzy environment. Acknowledgments I am grateful to the anonymous referees for their insightful suggestions. References [1] J. Adamek, W. Wechler, Minimization of R-fuzzy automata, in: Studien zur Algebra und Ihre Anwendungen, AkademieVerlag, Berlin, 1976.
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[2] P. Asveld, A bibliography on fuzzy automata, grammars and languages, Bull. EATCS 58 (1996) 187–196. [3] J. Berstel, C. Retenauer, Rational series and their languages, EATCS Monographs in Theoretical Computer Science, vol. 12, Springer, Berlin, 1988. [4] J. Brunner, W. Wechler, On behaviour of R-fuzzy automata, LNCS 45 (1976) 210–215. [5] K. Culik II, J. Karhumäki, Finite automata computing real functions, SIAM J. Comput. 23 (4) (1994) 789–814. [6] M. Droste, D. Kuske, Skew and infinitary formal power series, LNCS 2719 (2003) 426–438. [7] M. Droste, D. Kuske, Skew and infinitary formal power series, Technical Report 2003-38, Department of Mathematics and Computer Science, University of Leicester. [8] S. Eilenberg, Automata, Languages and Machines, vol. A, Academic Press, New York, 1974. [9] K. Krithivasan, K. Sharda, Fuzzy -automata, Inform. Sci. 138 (2001) 257–281. [10] W. Kuich, Semirings and formal power series: their relevance to formal languages and automata, Handbook of Formal Languages, vol. 1, Springer, Berlin, 1997, pp. 609–677. [11] W. Kuich, A. Salomaa, Semirings, Automata, Languages, EATCS Monographs in Theoretical Computer Science, vol. 5, Springer, Berlin, 1986. [12] L. Lee, L. Zadeh, Fuzzy languages and their acceptance by automata, in: Fourth Princeton Conference Information Sciences and Systems, 1970. [13] J. Mordeson, D. Malik, Fuzzy Automata and Languages: Theory and Applications, Chapman & Hall/CRC Boca Raton, London, New York, Washington, DC, 2002. [14] K. Peeva, Behaviour, reduction and minimization of finite L-automata, Fuzzy Sets and Systems 28 (2) (1988) 171–181. [15] K. Peeva, Fuzzy acceptors for syntactic pattern recognition, Internat. J. Approx. Reason. 5 (3) (1991) 291–306. [16] K. Peeva, Equivalence, reduction and minimization of finite automata over semirings, Theoret. Comput. Sci. 88 (1991) 269–285. [17] K. Peeva, Finite L-fuzzy machines, Fuzzy Sets and Systems 141 (3) (2004) 415–437. [18] K. Peeva, Finite L-fuzzy acceptors, regular L-fuzzy grammars and syntactic pattern recognition, Internat. J. Uncertainty, Fuzziness and Knowledge-Based Systems 12 (1) (2004) 89–104. [19] A. Ray, B. Chatterjee, A. Majumdar, A formal power series approach to the construction of minimal fuzzy automata, Inform. Sci. 55 (1–5) (1991) 189–207. [20] E. Santos, Maximin automata, Inform. Control 13 (1968) 363–377. [21] E. Santos, Regular fuzzy expressions, in: Fuzzy Automata and Decision Processes, North-Holland, New York, 1977, pp. 169–175. [22] M. Schützenberger, On the definition of a family of automata, Inform. Control 4 (1961) 245–270. [23] W. Thomas, Automata on infinite objects, Handbook of Theoretical Computer Science, Elsevier Science Publ. B.V., Amsterdam, 1990, pp. 133–191. [24] V. Topencharov, K. Peeva, Equivalence reduction and minimization of finite fuzzy automata, J. Math. Anal. Appl. 84 (1981) 270–281. [25] W. Wechler, R-fuzzy Automata with a time-variant structure, LNCS 28 (1975) 73–76. [26] E. Wechler, R-fuzzy grammars, LNCS 32 (1975) 450–456. [27] W. Wechler, Families of L-fuzzy languages, LNCS 56 (1977) 177–186. [28] W. Wechler, The Concept of Fuzziness in Automata and Language Theory, Akademie Verlag, Berlin, 1978.
Infinite fuzzy computations George Rahonis∗ Department of Mathematics, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece Received 19 May 2004; received in revised form 8 December 2004; accepted 12 December 2004 Available online 6 January 2005
Abstract We consider Büchi automata weighted over the fuzzy semiring F = ([0, 1], max, min, 0, 1). A Büchi theorem for the class of -recognizable series with coefficients in F, i.e., for behaviors of Büchi automata over F is established. Furthermore, we show that an infinitary language is Büchi recognizable iff its characteristic series is -recognizable. © 2004 Elsevier B.V. All rights reserved. Keywords: Fuzzy system models; Formal series; Infinite words
1. Introduction In 1961, Schützenberger [22] introduced the concept of a weighted automaton which is in fact a classical automaton M (over an alphabet ) whose transitions are equipped with a weight over a semiring S = (S, ⊕, , 0, 1). Then to each input word u and each run of M at u, a value from S is assigned which is induced by multiplying the weights of the transitions consisting the run. Furthermore, the weight of u is obtained by summing up the weights of all successful runs of M on u. In this way the behavior of M is a formal power series |M| : ∗ → S which is called recognizable. The classical Schützenberger theorem states that Re c(, S) = Rat(, S), ∗ Tel.: +30 2310 998330; fax: +30 2310 997983.
E-mail address: [email protected] (G. Rahonis). 0165-0114/$ - see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2004.12.003
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i.e., the class of recognizable formal power series over and S equals that of rational power series. Since then a large number of papers appeared in the literature, concerning weighted automata over several semirings, and formal power series contributed to many applications in Computer Science (for extended lists of references see [3,8,11,10]). On the other hand, automata over infinite objects can serve as models of infinite processes like operating systems, web servers applications, etc. Furthermore, they play a crucial role in monadic second-order logic and game theory. One of the most important machines over infinite words is the Büchi automaton. Intuitively, a Büchi automaton A over an alphabet accepts a finite word u in the way a classical automaton does, whereas it accepts an infinite word w if there exists an infinite run of A on w which starts at the initial state and meets at least one -final state infinitely often. A classical result of Büchi states that − Re c() = − Rat(),
i.e., the classes of all Büchi recognizable and -rational languages over coincide. In the last few years formal power series and weighted automata over infinite words have attracted the interest of researchers. In this case, convergence problems may arise according to the operations of the underlying semiring. The first effort to this direction has been made in [5]. In that paper Culik II and Karhumäki considered weighted automata over R (the reals) computing real functions. Since convergence problems arose in R, the authors restricted themselves to the special class of level automata. Krithivisan and Sharda introduced in [9] fuzzy -automata considering various modes of acceptance. Their fuzzy -automata with Büchi condition are in fact fuzzy Büchi automata in our sense, restricted on infinite words. Recently in [6] (see also [7]), a Büchi theorem is proved for infinitary power series over the semiring Rmax = (R 0 ∪ {−∞}, max, +, −∞, 0). In that case, convergence handicaps are dealt with by considering the skew product of power series instead of the classical one. In this paper we deal with formal power series over infinite words with coefficients in the fuzzy semiring F = ([0, 1], max, min, 0, 1). The -rational operations among infinitary series are max −, min −, ∗ −, and -closure. − Rat(, F ) stands for the class of -rational series over an alphabet with coefficients in F . We introduce the notion of fuzzy Büchi automata which are Büchi automata weighted over F . Our main result, an extension of Büchi theorem, states that − Re c(, F ) = − Rat(, F ),
where − Re c(, F ) is the set of series obtained as behaviors of fuzzy Büchi automata. A similar result for finite words has been proved by Santos in [21]. In that paper, the author used the so-called maximin automata [20] which are in fact weighted automata over F . In the framework of formal series theory, the result of Santos is a classical Schützenberger theorem [22]. In our paper, no convergence shortcomings appear within the fuzzy semiring, due to the max and min operations. The notion of the fuzzy Büchi automaton is the natural extension of the classical one, in the framework of formal series over F . Precisely, a Büchi automaton is a fuzzy Büchi automaton with all its weights equal to 1. The structure of the paper is the following: In Section 2, we present some preliminary matter concerning finitary series over F . We restate the result of Santos [21] in terms of formal series theory. Section 3 is devoted to infinitary series. We introduce the notion of fuzzy Büchi automata and we state an effective
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normalization procedure. We establish the Büchi theorem for the class of -recognizable series, i.e., series obtained as behavior of fuzzy Büchi automata. Next, we show that -recognizability is preserved by strict homomorphisms. Finally, we prove that a language is Büchi recognizable iff its characteristic series is -recognizable. 2. Finitary formal series A semiring A = (A, +, ·, 0, 1) consists of a set A endowed with two binary operations + and · and two constant elements 0 and 1, such that: (i) (ii) (iii) (iv)
(A, +, 0) is a commutative monoid, (A, ·, 1) is a monoid, a · (b + c) = a · b + a · c and (a + b) · c = a · c + b · c, for each a, b, c ∈ A (distributivity law), a · 0 = 0 = 0 · a for each a ∈ A.
Further A is called commutative if a · b = b · a for all a, b ∈ A. The interval F = [0, 1] with the max and min operations obtains the semiring structure. We call this semiring the fuzzy semiring (this name first appeared in [19]), and we denote it by F , i.e. F = (F, max, min, 0, 1).
Obviously F is commutative. In the sequel, we concentrate on infinite subsets of F ; thus, we use sup (least upper bound) instead of max, and inf (greatest lower bound) instead of min. We shall denote by sup A (resp. inf A) the least upper bound (resp. the greatest lower bound) of a set A. will stand for a finite alphabet and ∗ for the free monoid generated by , i.e., for the set of all finite words over . Let w = x0 x1 . . . xn−1 ∈ ∗ , with x0 , . . . , xn−1 ∈ . We shall use the notation w = w(0)w(1) . . . w(n − 1) where w(i) = xi , for i = 0, . . . , n − 1. will denote the empty word. Then + = ∗ − {}. A mapping S : ∗ → F is called formal power series over and F . The values S(w), w ∈ ∗ are usually written as (S, w) and referred to as the coefficients of the series. F ∗ denotes the set of all series over with coefficients in F . It is well-known that F ∗ inherits the semiring structure [3,11,10]. For S, T ∈ F ∗ the addition S ⊕ T : ∗ → F is defined by (S ⊕ T , w) = sup{(S, w), (T , w)},
for w ∈ ∗ ,
whereas the multiplication S T : ∗ → F is determined by (S T , w) = sup{inf{(S, w1 ), (T , w2 )}/w = w1 w2 },
for w ∈ ∗ .
The neutral elements for the above operations are, respectively, 0 : ∗ → F, and ∗
1 : → F,
for w ∈ ∗
(0, w) = 0 (1, w) =
1 0
if w = , else.
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The support of a series S ∈ F ∗ is the language sup p(S) = {w | w ∈ ∗ and (S, w) = 0}. A series with finite support is called polynomial. As usual F ∗ denotes the set of all polynomials over and F . Let a ∈ F and S ∈ F ∗ . The scalar product of a with S is the series aS : ∗ → F given by (aS, w) = inf{a, (S, w)}
for each w ∈ ∗ .
Then a series S : ∗ → F can be written as a formal sum S = sup{(S, w)w | w ∈ ∗ }, where for each w ∈ ∗ the series w : ∗ → F is defined by 1 if u = w, (w, u) = 0 else S will be called proper if (S, ) = 0. For S ∈ F ∗ and n 0, the consecutive powers S n of S are inductively determined by S 0 = 1,
S n+1 = S n S,
n 0.
It is clear that for n 1 and w ∈ ∗ , it holds (S n , w) = sup{inf{(S, w1 ), . . . , (S, wn )} | w = w1 . . . wn }. Further for S proper, the series S ∗ : ∗ → F is defined to be S ∗ = sup S n , i.e., (S ∗ , w) = sup{(S n , w) | n 0}. Obviously, (S n , w) = 0, for n > |w|, where |w| denotes the length of the word w. The rational operations of formal power series are ⊕−, − and ∗ −. Rat(, F ) is the least set of series in F ∗ containing the polynomials and closed under the rational operations. The elements of Rat(, F ) are named rational series. Remark. The reader may observe that we should define the ∗ −operation (as well as the -operation later) on each series, because of the completeness property of the fuzzy semiring (see [10]). Nevertheless, we restrict ourselves to proper series for technical reasons only (see Lemma 6 below). Obviously, this fact does not lighten our Büchi theorem. We now give the definition of fuzzy weighted automata. They are fuzzy automata weighted over F . In [20,21] such automata are called maximin automata. Further, fuzzy automata and fuzzy automata weighted over semirings have been extensively studied by many authors (see for e.g. [1,2,4, 12–18,24–28]).
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Definition 1. A fuzzy weighted automaton is a system A = (Q, , , f, init, ter) where • • • • • •
Q is the finite set of states, is the finite alphabet of inputs, ⊆ Q × × Q is the set of transitions, f : → F is a mapping assigning to each transition a weight from F,
init : Q → F is the initial distribution, and ter : Q → F is the terminal distribution.
Let w = w(0) . . . w(n − 1) ∈ ∗ . A run of A at w is a word rw ∈ Q∗ , rw = rw (0) . . . rw (n), rw (0), . . . , rw (n) ∈ Q, such that (rw (i), w(i), rw (i + 1)) ∈ , for each i = 0, . . . , n − 1. The weight of the run rw is inf{f (rw (i), w(i), rw (i + 1)) | i = 0, . . . , n − 1}. We define that the weight of the empty run i.e., the run on the empty word, equals 1. The behavior of A is the series |A| : ∗ → F determined by (|A|, w) = sup{inf{init(rw (0)), inf{f (rw (i), w(i), rw (i + 1)) | i = 0, . . . , n − 1}, ter(rw (n))} | rw is a run of A at w} for all w ∈ ∗ . Observe that (|A|, ) = 0 if there exists at least one state q ∈ Q, such that init(q) = 0 and ter(q) = 0. Since then, we assume that there exists an empty run r from q to q, with weight inf{init(q), ter(q)}. A formal power series S ∈ F ∗ is called recognizable if there exists a fuzzy weighted automaton A, with S = |A|. The set of all recognizable series in F ∗ is denoted by Re c(, F ). Theorem 2 (Santos [21], Schützenberger [22]). Re c(, F ) = Rat(, F ). 3. Infinitary series For a finite alphabet , we denote by the set of all infinite (or -words) over . An infinite word w ∈ is written in the form w = w(0)w(1)w(2) . . . , with w(i) ∈ , i = 0, 1, 2, . . . . The set of all words over is ∞ , i.e., ∞ = ∗ ∪ . The concatenation of two words in ∞ is defined in the following way: w1 w2 if w1 ∈ ∗ w1 w2 = w1 if w1 ∈ . An infinitary series over and F is a mapping S : ∞ → F . The set of all infinitary series over and F is denoted by F ∞ .
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Example 3. Let L ⊆ ∞ . The characteristic series of L, char L : ∞ → F is given by 1 if w ∈ L, (char L , w) = 0 else. A series S ∈ F ∞ can be written as a “sum" S = Sfin ⊕ Sinf with Sfin : ∗ → F and Sinf : → F , given, respectively, by (S, w), w ∈ ∗ , (Sfin , w) = 0 else and
(Sinf , w) =
(S, w), 0
w ∈ , else.
Consider S, T ∈ F ∞ . Their sum and product are the infinitary series S ⊕ T , S T whose coefficients are determined, respectively, by the following clauses: (S ⊕ T , w) = sup{(S, w), (T , w)}
for each w ∈ ∞ ,
and (S T , w) = sup{inf{(S, w1 ), (T , w2 )} | w = w1 w2 }
for each w ∈ ∗ ,
whereas (S T , w) = sup{(S, w), sup{inf{(S, w1 ), (T , w2 )} | w = w1 w2 , w1 ∈ ∗ }},
for each w ∈ .
Further the -closure of a proper finitary series S is the infinitary series S : → F with coefficients (S , w) = sup{inf{(S, w1 ), (S, w2 ), . . .} | w = w1 w2 . . . , with w1 , w2 , . . . ∈ ∗ }, for each w ∈ . The -rational operations in F ∞ are ⊕−, −, ∗ − and -closure, where ∗ − and -closure are defined for proper finitary series. − Rat(, F ) stands for the least class of series in F ∞ containing the polynomials and closed under the -rational operations. Its elements are termed -rational series. In the sequel, we introduce the notion of fuzzy Büchi automata. They are Büchi automata weighted over F . Such automata restricted on infinite words are called fuzzy -automata with Büchi condition in [9]. Definition 4. A fuzzy Büchi automaton (FBA) is a 7-tuple A = (Q, , , f, init, ter, ter ) where • Q is the finite state set, • is the finite input alphabet,
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• • • • •
281
⊆ Q × × Q is the set of transitions, f : → F is a mapping assigning to each transition a weight from F, init : Q → F is the initial distribution, ter : Q → F is the terminal distribution, and ter : Q → F is the -terminal distribution.
Consider a word w = w(0) . . . w(n−1) ∈ ∗ . A run of A at w is a word rw ∈ Q∗ , rw = rw (0) . . . rw (n), rw (0), . . . , rw (n) ∈ Q, such that (rw (i), w(i), rw (i + 1)) ∈ , for each i = 0, . . . , n − 1. The weight of the run rw is inf{f (rw (i), w(i), rw (i + 1)) | i = 0, . . . , n − 1}. The finitary behavior of A is the series |A|fin : ∗ → F with coefficients (|A|fin , w) = sup{inf{init(rw (0)), inf{f (rw (i), w(i), rw (i + 1)) | i = 0, . . . , n − 1}, ter(rw (n))} | rw is a run of A at w} for all w ∈ ∗ . Observe that the finitary behavior of A is the behavior of the fuzzy weighted automaton (Q, , , f, init, ter). Now let w = w(0)w(1) . . . be an infinite word over . A run of A at w is an infinite word rw = rw (0)rw (1) . . . ∈ Q such that (rw (i), w(i), rw (i + 1)) ∈ , for all i = 0, 1, . . . . Assume J ⊆ N to be an infinite index set (where N is the set of naturals). The weight of the run rw with terminals J is defined to be the number rwJ = inf{init(rw (0)), inf{f (rw (i), w(i), rw (i + 1)) | i = 0, 1, . . .}, inf{ter (rw (j )) | j ∈ J }}. The infinitary behavior of A is the series |A|inf : → F given by (|A|inf , w) = sup{rwJ | rw is a run of A at w, J ⊆ N is infinite} for each w ∈ . The behavior of A is the series | A | : ∞ → F defined by |A| = |A|fin ⊕ |A|inf . A series S ∈ F ∞ is termed to be -recognizable if there exists an FBA A, such that S = |A|. We denote by − Re c(, F ) the class of all -recognizable series over the alphabet .
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Definition 5. An FBA A = (Q, , , f, init, ter, ter ) is called normalized if there are two states q0 , qt ∈ Q such that 1 if q = qt , 1 if q = q0 , ter(q) = init(q) = 0 else, 0 else and there are no transitions of the forms (q, a, q0 ) and (qt , a, q) in . We call q0 and qt the initial and terminal state, respectively. Such an automaton will be denoted by A = (Q, , , f, q0 , qt , ter ). Further, an FBA A = (Q, , , f, init, ter, ter ) is termed weak normalized if it has an initial state q0 as above. In this case we write A = (Q, , , f, q0 , ter, ter ). We establish the following lemma. Lemma 6. Let A = (Q, , , f, init, ter, ter ) be an FBA. We can effectively construct a normalized FBA A such that sup p(|A |fin ) = sup p(|A|fin ) ∩ + and |A |inf = |A|inf . Further, we can effectively construct a weak normalized FBA A such that |A | = |A|. Proof. Consider the new states q0 , qt and put Q = Q ∪ {q0 , qt }. Let A = (Q , , , f , q0 , qt , ter ) be the normalized FBA with set of transitions = ∪ {(q0 , a, q ) | ∃q ∈ Q : (q, a, q ) ∈ } ∪ {(q, a, qt ) | ∃q ∈ Q : (q, a, q ) ∈ } ∪ {(q0 , a, qt ) | ∃q, q ∈ Q : (q, a, q ) ∈ }.
The weight function f : → F is given by the following clauses: • • • •
f (q, a, q ) = f (q, a, q ), ∀(q, a, q ) ∈ , f (q0 , a, q ) = sup{inf{init(q), f (q, a, q )} | (q, a, q ) ∈ }, ∈ , q ∈ Q, f (q, a, qt ) = sup{inf{f (q, a, q ), ter(q )} | (q, a, q ) ∈ }, ∈ , q ∈ Q, f (q0 , a, qt ) = sup{inf{init(q), f (q, a, q ), ter(q )} | (q, a, q ) ∈ }, ∈ .
The reader will have no difficulties in verifying that the FBA A has the announced properties. The weak normalization procedure of A is similar to the previous one. Now we are ready to prove our main result. Theorem 7. A series S ∈ F ∞ is -rational iff it is -recognizable. Proof. We establish the proof in two steps. Step 1. − Rat(, F ) ⊆ − Re c(, F ). It suffices to prove that the class − Re c(, F ) contains the polynomials and it is closed under the -rational operations. Because of Theorem 2, Rat(, F ) ⊆ − Re c(, F ) and thus the polynomials are -recognizable. Let S ∈ − Re c(, F ) be a proper finitary series i.e. S ∈ Rec(, F ). Then once again by Theorem 2, S ∗ ∈ Re c(, F ), and so S ∗ ∈ − Re c(, F ). Further, there exists a normalized FBA A =
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283
(Q, , , f, q0 , qt , 0) such that |A| = S. We identify the states q0 and qt , and we consider the FBA A = (Q, , , f, q0 , 0, ter )
with ter (q)
=
1 0
if q = q0 , else.
Let w = w(0)w(1) . . . be an infinite word in , rw be a run of A at w, and J ⊆ N an infinite set, namely J = {j1 , j2 , . . .}, with j1 < j2 < · · · . Then rwJ = inf{init(rw (0)), inf{f (rw (i), w(i), rw (i + 1)) | i = 0, 1, . . .}, inf{ter (rw (j )) | j ∈ J }}. Assume that rwJ = 0. Then rw (0) = q0 and rw (j ) = q0 , ∀j ∈ J . Thus rwJ = inf{inf{inf{f (rw (i), w(i), rw (i + 1)) | i = jk , jk + 1, . . . , jk+1 − 1} | k = 0, 1, . . .}, inf{ter (rw (jk )) | k = 0, 1, . . .}} = inf{inf{f (rw (i), w(i), rw (i + 1)) | i = jk , jk + 1, . . . , jk+1 − 1} | k = 0, 1, . . .}, where j0 = 0. That is rwJ = inf{(S, w(0) . . . w(j1 − 1)), (S, w(j1 ) . . . w(j2 − 1)), . . .}. Conversely, by similar arguments we can show that for each decomposition w = w1 w2 . . . , w1 , w2 . . . ∈
∗ of w ∈ , we can find an infinite J ⊆ N and a run rw of A on w, such that
rwJ = inf{(S, w1 ), (S, w2 ), . . .}. We conclude that |A | = S and so S ∈ − Re c(, F ). Now let S, T ∈ − Re c(, F ) and A1 = (Q1 , , 1 , f1 , init 1 , ter 1 , ter 1 ), A2 = (Q2 , , 2 , f2 , init 2 , ter 2 , ter 2 ) be two FBA’s with |A1 | = S and |A2 | = T . Without any loss, we can assume that Q1 ∩ Q2 = ⭋. We construct the FBA A = (Q, , , f, init, ter, ter )
whose components are given by • Q = Q1 ∪ Q2, = 1 ∪ 2 , f1 (q, a, q ) if (q, a, q ) ∈ 1 , • f (q, a, q ) = f2 (q, a, q ) if (q, a, q ) ∈ 2 , init 1 (q) if q ∈ Q1 , • init(q) = init 2 (q) if q ∈ Q2 , ter 1 (q) if q ∈ Q1 , • ter(q) = ter (q) if q ∈ Q2 , 2 ter 1 (q) if q ∈ Q1 , • ter (q) = ter 2 (q) if q ∈ Q2 . It is a tedious task to formally state that |A| = S ⊕ T .
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Further, let S, T ∈ − Re c(, F ) with sup p(S) ⊆ ∗ and sup p(T ) ⊆ . Let A1 = (Q1 , , 1 , f1 , q01 , qt1 , 0) be a normalized FBA with |A1 | = S, and A2 = (Q2 , , 2 , f2 , q02 , 0, ter 2 ) be a weak normalized FBA with |A2 | = T . Again we assume that Q1 ∩ Q2 = ∅. We consider the FBA, A = (Q, , , f, q01 , 0, ter 2 ),
where Q, and f are determined as in the above construction, and we have identified the states qt1 and q02 . The reader will have no difficulties in showing that |A| = S T . In order to complete the proof of Step 1, we have to establish that S T ∈ − Re c(, F ) for arbitrary -recognizable series S, T . But S T = (Sfin ⊕ Sinf ) (Tfin ⊕ Tinf ) and because of the distributivity law in F S T = (Sfin Tfin ) ⊕ (Sfin Tinf ) ⊕ (Sinf Tfin ) ⊕ (Sinf Tinf ). By the definition of the -operation we have (Sinf Tfin ) = (Sinf Tinf ) = Sinf and thus S T = (Sfin Tfin ) ⊕ (Sfin Tinf ) ⊕ Sinf which by the previous arguments gives S T ∈ − Re c(, F ) as wanted. Step 2. − Re c(, F ) ⊆ − Rat(, F ). Let S ∈ − Re c(, F ). Without any loss we can assume S to be proper. There exists a normalized FBA A = (Q, , , f, q0 , qt , ter ) with |A| = S. Consider the FBA’s, A1 = (Q, , , f, q0 , qt , 0) (obviously |A1 | = |A1 |fin = Sfin ), A2 = (Q, , , f, q0 , ter 2 , 0) with ter 2 = ter , A3 = (Q, , , f, init 3 , ter 3 , 0) with init 3 = ter 3 = ter .
It is not hard to understand that the support of A2 is composed of all the finite prefixes of infinite words in the support of A. Further, Sinf = |A2 | |A3 | = |A2 |fin (|A3 |fin ) . Thus, S = |A1 |fin ⊕ (|A2 |fin (|A3 |fin ) ). Since |A1 |fin , |A2 |fin , |A3 |fin are rational power series (Theorem 2) S is -rational and this completes the proof. Next we show that strict homomorphisms preserve -recognizability.
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285
Let , be finite alphabets and h : ∗ → ∗ be a homomorphism. Recall that h is called strict if h(a) ∈ + , for all a ∈ . h is extended to infinite words in the obvious way: If w = w(0)w(1)... ∈ , then h(w) = h(w(0))h(w(1)).... Furthermore, h is extended to a mapping −
h : F ∞ → F ∞
by the formula −
h(S) = sup{(S, w)h(w) | w ∈ ∞ }
for all S ∈ F ∞ .
Proposition 8. Assume S ∈ F ∞ to be -recognizable and h : ∗ → ∗ to be a strict homomor−
phism. Then the series h(S) ∈ − Re c(, F ). Proof. Let A = (Q, , , f, q0 , ter, ter ) be a weak normalized FBA with behavior S. For each a ∈ , such that h(a) = 1 . . . n , i ∈ , i = 1, . . . , n, and each transition (q, a, q ) ∈ , we consider the new states p1 , . . . , pn−1 not belonging to Q, and the new transitions (q, 1 , p1 ), (p1 , 2 , p2 ), . . . , (pn−1 , n , q ). Let P be the set of the new states obtained in the above way, and the set of the new transitions. Let also A = (Q , , , f , q0 , ter , ter )
be the weak normalized FBA with its components given by the following clauses: • Q = Q ∪ P , • f (q, 1 , p1 ) = f (q, a, q ) and f (p1 , 2 , p2 ) = · · · = f (pn−1 , n , q ) = 1, where the transitions (q, 1 , p1 ),(p1 , 2 , p2 ), . . . , (pn−1 , n , q ) obtained from (q, a, q ) in the above-described procedure, ter(q) if q ∈ Q, • ter (q) = 0 else, ter (q) if q ∈ Q, • ter (q) = 0 else. Assume now that w = w(0) . . . w(n − 1) ∈ ∗ and rw is a run of A at w. Obviously, there is a run rh(w) of A at h(w), with the same weight. Conversely, if ru is a run of A at u ∈ ∗ , then there exists a run rw of A at the word w, such that h(w) = u, and the weights of ru and rw are equal. This means that −
h(S)fin = |A |fin . Further, let w ∈ , rw be a run of A at w and J ⊆ N be an infinite index set such that rwJ = 0. From the construction of the automaton A , there is a run rh(w) of A at h(w) and an infinite index set J ⊆ N
J . On the other hand, if r is a run of A at u ∈ , and J ⊆ N is an infinite index such that rwJ = rh(w) u
set such that ruJ = 0, then there clearly exists a word w ∈ , with h(w) = u, a run rw of A at w, and
−
−
an infinite index set J ⊆ N such that rwJ = ruJ . Thus, h(S)inf = |A |inf . We conclude that h(S) = |A |, −
and so h(S) ∈ − Re c(, F ).
In the sequel, we investigate the relation among Büchi recognizable languages and -recognizable series. First, we recall the notion of a Büchi recognizable language [22].
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Definition 9. A Büchi automaton is a system A = (Q, , , q0 , T , T ), where • • • • • •
Q is the finite state set, is the input alphabet,
q0 ∈ Q is the initial state, ⊆ Q × × Q is the set of transitions, T ⊆ Q is the set of final states, and T ⊆ Q is the set of -final states.
A run of A on a finite word w = w(0) . . . w(n − 1) ∈ ∗ is a word rw = rw (0) . . . rw (n) ∈ Q∗ such that (rw (i), w(i), rw (i + 1)) ∈ , for each i = 0, 1, . . . , n − 1. rw is called successful if rw (0) = q0 and rw (n) ∈ T . A run of A on an infinite word w = w(0)w(1) . . . ∈ is an infinite word rw = rw (0)rw (1) . . . ∈ Q , such that (rw (i), w(i), rw (i + 1)) ∈ , for all i = 0, 1, . . . . An infinite run rw is called successful if rw (0) = q0 and at least one state of T occurs infinitely many often among the rw (i)’s. The language accepted by A is L(A) = {w | w ∈ ∞ , there exists a successful run of A at w}. L ⊆ ∞ is called Büchi recognizable if L = L(A) for some Büchi automaton A. First, we prove that Proposition 10. Let L ⊆ ∞ be a Büchi recognizable language. Then its characteristic series char L : ∞ → F is -recognizable. Proof. Assume A = (Q, , , q0 , T , T ) to be a Büchi automaton recognizing L. Consider the weak normalized FBA, −
A = (Q, , , f, q0 , ter, ter )
with f : → F given by f (q, a, q ) = 1, and
ter(q) =
1 0
for each (q, a, q ) ∈
if q ∈ T , else,
ter (q) =
1 0
if q ∈ T , else.
It is clear from the constructions that for a given word w ∈ ∞ a successful run of A at w exists iff
−
−
(| A |, w) = 1. Thus | A | = char L . We obtain that char L ∈ − Re c(, F ). Next we establish the inverse implication. Proposition 11. Let L ⊆ ∞ . If char L ∈ − Re c(, F ) then L is Büchi recognizable.
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Proof. Let A = (Q, , , f, q0 , ter, ter ) be a weak normalized FBA with behavior char L , i.e. w∈L
iff (|A|, w) = 1.
We construct the Büchi automaton −
A = (Q, , , q0 , T , T )
with • T = {q | q ∈ Q and ter(q) = 0}, and • T = {q | q ∈ Q and ter (q) = 0}. To each run rw of A on a word w ∈ ∞ , corresponds a run r¯ of A¯ on w and vice versa (in fact r¯ = rw ). If w = w(0) . . . w(n − 1) ∈ ∗ , then inf{init(rw (0)), inf{f (rw (i), w(i), rw (i + 1) | i = 0, . . . , n − 1}, ter(rw (n))} = 0 iff r¯ is a successful run of A¯ . Further, if w ∈ , and J ⊆ N is an infinite index set such that rwJ = 0, then r¯w is successful. The converse is also obvious. We conclude (|A|, w) = 1
iff w ∈ L(A¯ ),
i.e., (char L , w) = 1
iff w ∈ L(A¯ )
and so L = L(A¯ ) which means that L is a Büchi recognizable language. 4. Conclusion We have considered Büchi automata weighted over the fuzzy semiring and we established a normalization procedure for them. We showed that their behaviors called -recognizable series constitute the class of -rational series over F . Furthermore, we proved that a string language is Büchi recognizable iff its characteristic series is -recognizable. Hopefully, Büchi automata over F can stand as models of infinite processes under a fuzzy environment. Acknowledgments I am grateful to the anonymous referees for their insightful suggestions. References [1] J. Adamek, W. Wechler, Minimization of R-fuzzy automata, in: Studien zur Algebra und Ihre Anwendungen, AkademieVerlag, Berlin, 1976.
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